Brian Bolt has written numerous resource books for teachers containing collections of rich mathematical problems, puzzles, investigations and games. Some are descriptions of classic problems and puzzles while others are new creations. I think these books are a great resource and I wanted to share three of my favourite problems from them.

Number the Sectors

Find numbers A, B, C, D, E, and F for the six sectors so that the number in a sector, or the total of the numbers in a set of adjacent sectors, gives all the integers from 1 to 25 inclusive.

Is it possible to obtain a larger range of numbers in this way? Investigate the numbers required to give the largest range of totals when the circle is divided into 2, 3, 4 5, ..., n sectors.

This is problem #53 from Even More Mathematical Activities (1987) and problem #72 fromThe Mathematical Funfair (1989). Instead of starting by asking students to find a set of numbers that gives all the integers from 1 to 25, I like to create an example as a class and then challenge them to do better (get to a larger number). You can also ask them to prove what the maximum value is.

Bolt has an alternate version of this puzzle in A Mathematical Pandora's Box (1993) (#12 Can you Do Better), which has 5 sectors around a central circle. This version can be found is online at NRICH Maths as the Number Daisy.

How Large a Number Can You Make?

Make the largest number with just the digits 1, 2, and 3 once only and any mathematical symbols you are aware of, but no symbol is to be used more than once. The challenge is to see who can make the largest number. Here are some numbers to get the ball rolling:

This is problem #83 from Even More Mathematical Activities, (1987). I've given this as a warm-up problem for high school students and this often leads to a discussion of how to know which is bigger, 2^31 or 3^21?

Make a Century

By putting arithmetical signs in suitable places between the digits make the following sum correct: 1 2 3 4 5 6 7 8 9 = 100There is more than one solution. See how many you can find.

This is problem #127 from Mathematical Activities (1982). I would start this challenge with students by asking them to make an expression using the numbers from 1 to 9 to make a value as close as possible to 100. I would then add on the challenge to try to find an expression exactly equal to 100. There is a very similar problem called Make 100 on NRICH Maths.

I saw an earlier version of this as problem #94 in Amusements in Mathematics (1917) by Henry Ernest Dudeney. In Dudeney's version, he includes an additional challenge to try to find a solution which "employs (1) the fewest possible signs, and (2) the fewest possible separate strokes or dots of the pen. That is, it is necessary to use as few signs as possible, and those signs should be of the simplest form. The signs of addition and multiplication (+ and ×) will thus count as two strokes, the sign of subtraction (-) as one stroke, the sign of division (÷) as three, and so on."

What are Your Favourite Problems?

Do you have a favourite problem or puzzle from one of Brian Bolt's puzzle books? Do you have other favourite collections of puzzles?

Being on Twitter and following hashtags like #MTBoS and #ITeachMath allows me to see classroom mathematics well beyond my physical horizons. I get to glimpse creative and engaging mathematics education around the globe. Recently I saw a couple of different ideas that I've tried to adapt and apply for myself.

Mysteries

Math mysteries are an idea I saw posted on Twitter by Richard Perring (@LearningMaths). He shared a math mystery he created for completing the square. Many more of these activities are printed in his book Talking Maths from The Association of Teachers of Mathematics (ATM). The goal of the activity is to follow a set of clues in order to fill a 3 x 3 grid with the correct expressions or equations. There is a lot of thinking to be done in order to determine the correct placement of each expression. The puzzle like quality of these activities make them more engaging and purposeful.

Treasure Hunt

I was looking for something completely different when I ran across a Treasure Hunt Math Activity on TES. This activity was created and posted by @colmanweb. It is a series of problems with corresponding solutions. The solutions are placed on a treasure map and as each problem is solved, the solution is crossed off. Once all the problems are solved there should be one remaining number on the map that has not been crossed off. This is the location of the treasure. I made a version of this activity for integer addition and subtraction.

​I liked this idea because it is relatively easy to create; just a find a series of questions with unique answers. Also, students get instant feedback. If their answer isn't on the map, they know they've made a mistake. I would call this purposeful practice as there is a goal to achieve at the end of the activity. There is a reason to persevere. Once students are familiar with the activity, you could give them a blank template (or they could hand draw their own version) and they could work in small groups to make their own treasure hunt activity (and answer key) and share it with each other.

The Role of Practice

I recently read Mark Chubb's (@MarkChubb3) blog post on the role of practice in math class. He discussed the differences between "rote practice" and "dynamic practice". Rote practice involves following procedures, drill and repetition while dynamic practice involves active student thinking, playful experiences and puzzles. I think that the Mystery activity is a more "dynamic" activity than doing the Treasure Hunt activity. However, I think that creating your own Treasure Hunt activity does involve additional characteristics of dynamic practice.